Answer basic probability and statistics questions

You are given several short, independent probability and statistics questions similar to those in a data / ML screening test. Answer all sub-questions.

1. Ordering Poisson distributions by their rate parameter

You are told that three different Poisson-distributed random variables $X_A, X_B, X_C$ have their probability mass functions (PMFs) plotted on the same graph (support on non-negative integers). The plots are described as follows:

• Distribution A : Most of its probability mass is concentrated on values 0, 1, and 2. The mode (highest bar) is at 1. The probability at 0 is high, and probabilities drop off quickly after 3.
• Distribution B : The mode is at 3. The distribution is more spread out than A: there is still noticeable probability up to around 6 or 7, but very little after that.
• Distribution C : The mode is at 5. The distribution is the most spread-out of the three, with noticeable probability from around 2 up to 10 or more.

All three are Poisson distributions with parameters $\lambda_A$, $\lambda_B$, and $\lambda_C$ respectively.

Question 1: Based on the qualitative description of the plots, order the three rate parameters from smallest to largest:

• (a) $\lambda_A$ , $\lambda_B$ , $\lambda_C$
• (b) $\lambda_C$ , $\lambda_B$ , $\lambda_A$
• (c) $\lambda_A$ , $\lambda_C$ , $\lambda_B$
• (d) $\lambda_B$ , $\lambda_A$ , $\lambda_C$

Pick the correct ordering and briefly justify your choice.

2. Identifying type of bias in a study

A startup incubator wants to understand “what makes startups successful.” They collect data only from companies that have already raised Series C or later funding and are still operating. They analyze features such as team size, prior founder experience, average age of founders, and industry, and then publish a report claiming: “These are the characteristics that make startups successful.”

You are asked: What is the primary type of bias in this study? Choose the best option and briefly explain.

• (a) Survivor (survivorship) bias
• (b) Sampling bias (non-representative sample)
• (c) Recall bias
• (d) No bias; the study design is appropriate

(You may mention more than one kind of bias if relevant, but identify the primary/statistically standard name.)

3. True/False questions on covariance and correlation

For each of the following statements about covariance and correlation between two real-valued random variables $X$ and $Y$, answer True or False and provide a brief justification.

1. Statement A: If $\text{Cov}(X, Y) = 0$ , then $X$ and $Y$ are independent.
2. Statement B: The Pearson correlation coefficient $\rho_{XY}$ is always between $-1$ and $1$ , inclusive.
3. Statement C: The Pearson correlation coefficient between $X$ and $Y$ is given by

$\rho_{XY} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y},$

where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$, respectively. 4. Statement D: If we rescale $X$ by a positive constant $a > 0$, i.e., define $X' = aX$, then the correlation between $X'$ and $Y$ is the same as the correlation between $X$ and $Y$. 5. Statement E: A very high correlation between $X$ and $Y$ implies that $X$ causes changes in $Y$.

State True/False for each and justify in one or two sentences.